In kernel-based regression learning, optimizing each kernel individually is useful when the data density, curvature of regression surfaces (or decision boundaries) or magnitude of output noise (i.e., heteroscedasticity) varies spatially. Unfortunately, it presents a complex computational problem as the danger of overfitting is high and the individual optimization of every kernel in a learning system may be overly expensive due to the introduction of too many open learning parameters. Previous work has suggested gradient descent techniques or complex statistical hypothesis methods for local kernel shaping, typically requiring some amount of manual tuning of meta parameters. In this paper, we focus on nonparametric regression and introduce a Bayesian formulation that, with the help of variational approximations, results in an EM-like algorithm for simultaneous estimation of regression and kernel parameters. The algorithm is computationally efficient (suitable for large data sets), requires no sampling, automatically rejects outliers and has only one prior to be specified. It can be used for nonparametric regression with local polynomials or as a novel method to achieve nonstationary regression with Gaussian Processes. Our methods are particularly useful for learning control, where reliable estimation of local tangent planes is essential for adaptive controllers and reinforcement learning. We evaluate our methods on several synthetic data sets and on an actual robot which learns a task-level control law.

Local linear regression performs very well in many low-dimensional forecasting problems. In high-dimensional spaces, its performance typically decays due to the well-known "curse-of-dimensionality". A possible way to approach this problem is by varying the "shape" of the weighting kernel. In this work we suggest a new, data-driven method to estimating the optimal kernel shape. Experiments using an artificially generated data set and data from the UC Irvine repository show the benefits of kernel shaping. 1 Introduction Local linear regression has attracted considerable attention in both statistical and machine learning literature as a flexible tool for nonparametric regression analysis [Cle79, FG96, AMS97]. Like most statistical smoothing approaches, local modeling suffers from the so-called "curse-of-dimensionality", the well-known fact that the proportion of the training data that lie in a fixed-radius neighborhood of a point decreases to zero at an exponential rate with increasing dimension of the input space.

Local linear regression performs very well in many low-dimensional forecasting problems. In high-dimensional spaces, its performance typically decays due to the well-known "curse-of-dimensionality". A possible way to approach this problem is by varying the "shape" of the weighting kernel. In this work we suggest a new, data-driven method to estimating the optimal kernel shape. Experiments using an artificially generated data set and data from the UC Irvine repository show the benefits of kernel shaping. 1 Introduction Local linear regression has attracted considerable attention in both statistical and machine learning literature as a flexible tool for nonparametric regression analysis [Cle79, FG96, AMS97]. Like most statistical smoothing approaches, local modeling suffers from the so-called "curse-of-dimensionality", the well-known fact that the proportion of the training data that lie in a fixed-radius neighborhood of a point decreases to zero at an exponential rate with increasing dimension of the input space.

Local linear regression performs very well in many low-dimensional forecasting problems. In high-dimensional spaces, its performance typically decays due to the well-known "curse-of-dimensionality". A possible way to approach this problem is by varying the "shape" of the weighting kernel. In this work we suggest a new, data-driven method to estimating the optimal kernel shape. Experiments using anartificially generated data set and data from the UC Irvine repository show the benefits of kernel shaping. 1 Introduction Local linear regression has attracted considerable attention in both statistical and machine learning literature as a flexible tool for nonparametric regression analysis [Cle79, FG96, AMS97]. Like most statistical smoothing approaches, local modeling suffers from the so-called "curse-of-dimensionality", the well-known fact that the proportion of the training data that lie in a fixed-radius neighborhood of a point decreases to zero at an exponential rate with increasing dimension of the input space.